Model of crystallizer

Mathews and Rawlings (1998) successfully applied model-based control using solids hold-up and liquid density measurements to control the filtrability of a photochemical product. Togkalidou etal. (2001) report results of a factorial design approach to investigate relative effects of operating conditions on the filtration resistance of slurry produced in a semi-continuous batch crystallizer using various empirical chemometric methods. This method is proposed as an alternative approach to the development of first principle mathematical models of crystallization for application to non-ideal crystals shapes such as needles found in many pharmaceutical crystals. 

The relaxation of a thermodynamic system to an equilibrium configuration can be conveniently described by a master equation [47]. The probability of finding a system in a specific state increases by the incoming jump from adjacent states, and decreases by the outgoing jump from this state to the others. From now on we shall be specific for the lattice-gas model of crystal growth, described in the previous section. At the time t the system will be found in the state. S/ with a probability density t), and its evolution... 

J. Villain, A. Pimpinelli. Physique de la croissance cristalline. Paris Editions Eyrolles, 1995 (in French) J. Villain. Continuum models of crystal growth from atomic beams with and without desorption. J Phys (France) I 7 19, 1991. 

Nicholas, J. E., An Atlas of Models of Crystal Surfaces. Gordon Breach, New York, 1965. 

For dynamic experiments, such as modelling of crystal growth or circuit processing. 

J. Villain, Continuum models of crystal growth from atomic beams with and without desorption,... 

As explained above, three fundamental models of crystal growth mechanism were established in relation to the roughness of interfaces these are illustrated in Fig. 3.14. At present, there is no other known growth mechanism that is essentially different from these three. Therefore, we shall analyze the morphology of crystals, the main topic of this book, based on these three growth mechanisms. 

From Eq, (1) it is clear that a model of crystal polarization that is adequate for the description of the piezoelectric and pyroelectric properties of the P-phase of PVDF must include an accurate description of both the dipole moment of the repeat unit and the unit cell volume as functions of temperature and applied mechanical stress or strain. The dipole moment of the repeat unit includes contributions from the intrinsic polarity of chemical bonds (primarily carbon-fluorine) owing to differences in electron affinity, induced dipole moments owing to atomic and electronic polarizability, and attenuation owing to the thermal oscillations of the dipole. Previous modeling efforts have emphasized the importance of one more of these effects electronic polarizability based on continuum dielectric theory" or Lorentz field sums of dipole lattices" static, atomic level modeling of the intrinsic bond polarity" atomic level modeling of bond polarity and electronic and atomic polarizability in the absence of thermal motion. " The unit cell volume is responsive to the effects of temperature and stress and therefore requires a model based on an expression of the free energy of the crystal. 

Examples of crystals are shown in Eigs. 32-37 and in various other drawings in this book. Familiarity with crystal symmetry is, however, best attained by handling and contemplating idealized models of crystals. 

Modelling of crystallization was discussed in Section 2.8. Now, we shall develop a model for superimposed polymerization and crystallization processes. This model is important for calculating temperature evolution during reactive processing, because an increase in temperature, regardless of its cause, influences the kinetics of both polymerization and crystallization. This concept is expressed by the following equation for the rate of heat output from the superimposed proceses 102,103... 

Higgins MD (1994) Numerical modeling of crystal shapes in thin sections Estimation of crystal habit and true size. Am Mineral 79 113-119... 

Such an order is difficult to rationalize in terms of electrostatic energies embodied in the simple point-charge model of crystal field theory. For example, the charged O2- anion precedes the dipolar H20 molecule in the spectrochemical series. Note that the spectrochemical series differs from the nephelauxetic series discussed in chapter 11 (eq. 11.6), which is a measure of the degree of covalent bonding. 

The development of statistical theory of processes of phase transformations realized by hydrogenation of metalfulleride, the elucidation and substantiation of conditions of proceeding reaction (1) are of more direct interest to scientists. The atomic configuration model of crystals is used below without considering possible processes in subsystems of interstitial atoms as hydrogen and platinum. 

Teodosiu C. Elastic models of crystal defects. Springer, Heidelberg, 1982... 

There seems to be a remarkable consistency between Buckminster Fuller s evaluation of chemistry ( chemists consider volumes as material domains and not merely as abstractions, see Chapter 1— Introduction) and Kitaigorodskii s geometrical model of crystal structures. In one of his last statements Kitaigorodskii (Figure 9-54) said (when asked about his most important achievements in science) I ve shown that the molecule is a bodyd One can take it, one can hit with it it has mass, volume, form, hardness. I followed the ideas of Democritos... [100],... 

The interatomic bonds that produce the crystalhne structures of minerals are briefly discussed first. This is followed by general mles used in constmcting models of crystal structures of phosphate minerals, then the crystal structures of orthophosphate mineral forms. The discussion is brief because the emphasis of the book is on practical aspects of novel phosphate ceramics and cements. Readers interested in more details are referred to Corbridge et al. [1] and Kanazawa [2]. 

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